The King of Infinite Space: Euclid and His Elements

“The King of Infinite Space is for anyone who cares about Euclid, geometry, the philosophy of mathematics and, most especially, for those who appreciate fine writing.”

David Berlinski’s slim book The King of Infinite Space is not your typical biography.

Concerning Euclid and his book on geometry, the Elements, The King of Infinite Space is surprisingly compelling. Dr. Berlinski goes about his business in an artful manner, at times serious and straightforward, at times stream-of-consciousness or poetic. The spirit of this book is something this reviewer can only hope to capture.

Euclid was born in the fourth century BCE and died sometime in the third. Much of Euclid’s life we only know only indirectly through the writings of others—through copies of copies. His Elements comprises 13 books, the first four (of which have been the longest lived) concern geometry, the study of shapes in space—points, straight lines, circles, squares, right angles, triangles, rectangles, magnitudes, proportions, and solids. Modern versions of the Elements are based on a 10th century Greek text; even so, the Elements have been the most successful of all mathematical textbooks and the oldest complete text in the Western mathematical tradition.

Dr. Berlinksi claims that the Elements draws a historical line in the sand that separates the unorganized before from the organized after, and that Euclid also provides a double insight, the first that geometry could be organized into a whole, and second that geometric propositions could be, must be logical.

That Euclid provided conclusions was not enough; Euclid also provided proofs, chains of logic that connect axioms to conclusions. And if that were not enough, Dr. Berlinski also claims the Elements to be a great artistic achievement, which is “unusual as a mathematical treatise in that it is meticulously illustrated. For every theorem there is a picture.”

Euclid starts off his Elements with a handful of assumptions, called axioms and from these derives propositions and conclusions about geometry. From five axioms Euclid derived 467 theorems, forever accessible to anyone capable of following a logical argument. Before providing the reader Euclid’s axioms, Dr. Berlinski first gives us Euclid’s five “common notions,” or beliefs:

1. Things that are equal to the same thing are equal to one another.
2. If equals are added to equals, the wholes are equal.
3. If equals are subtracted from equals, the remainders are equal.
4. Things that coincide with one another are equal to one another.
5. The whole is greater than the part.

These beliefs are followed by Dr. Berlinksi’s comment, “It was Euclid’s business to say what those beliefs were. And our business to say what they mean.” Which he then proceeds to do but first points out that the greatest confusion of Euclid’s notions occurs in the gray space between the abstract and the concrete.

Earlier, Dr. Berlinski places Euclid into context alongside Plato, noting that the geometry of ancient Greece could not be separated from Plato’s ideal forms. But even Dr. Berlinski shows confusion in this gray space. Note his explanation of the word “coincide” in the fourth common notion: “If it is true that concrete triangles are never coincident, it is equally true that abstract triangles cannot be moved. They are beyond space and time.”

This reviewer, a computer programmer by profession, has no such difficulty imagining abstract of triangles moving—in fact, the more abstract, the easier to move in one’s mind.

Imagination is the enabler of transcending space and time, permitting the very conception of the abstract, let alone giving abstract triangles the freedom to move. This reviewer earlier today imagined seven impossible things (and all before breakfast). But that is just nitpicking.

Euclid’s five axioms are also provided. Here are the first four:

1. Between any two points there exists a unique straight line.
2. For any straight line segment, there exists a unique extension.
3. For every point, there exists a unique circle of fixed radius.
4. All right angles are equal.

Pay close attention to Dr. Berlinski before his presenting Euclid’s fifth axiom. “The fifth and final axiom of Euclid’s system is more famous than the other four. It is said to have troubled Euclid, who squirmed and turned, wheezed and whistled, before accepting it.” Now read the axiom for yourself:

5. If a straight line falling on two straight lines makes interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on the side on which are the angles less than two right angles.

Not only may a reader directly experience Dr. Berlinski’s wordplay in this quote but also share the deeper mathematical “Aha!” moment: that there is something significant (and unknown to Euclid himself) that causes this awkwardness.

There are more “aha” opportunities for the reader. Dr. Berlinski points out that in axiom four, Euclid assumes all right angles to be equal without defining what a right angle was (!), something that this reviewer stubbed his toe against in high school geometry.

The reader might also enjoy Dr. Berlinski’s metaphor. “It is not so much that Euclid’s definition is smudged. There seems to be no brass underneath the definition, no matter how much polish is applied to the surface.”

Axiom five for all its awkwardness provides mathematicians the property of flatness. Curved space was not imagined until the 19th century by Carl Friedrich Gauss. With curved space, the shortest distance between two lines might no longer be a straight line, for example on the surface of the Earth the shortest path is a great curve.

The Elements has 23 definitions for points, lines, and planes. A few of these are chosen by Dr. Berlinski for closer inspection, which he decorates with comments of later mathematicians including Newton, Descartes, Pascal, Weyl, Hardy, and Hilbert. Though few observations can match Dr. Berlinski’s own on the matter, “[t]hey reveal a great mind entering uncertainly into a space that logicians would not fully command for two thousand years. The definition are what they seem: an instruction, a guide to Euclid’s thoughts, a way into the labyrinth.”

Several of Euclid’s proofs are examined; those that the author believes are exceptional or provide illumination into Euclid’s thought processes, though Dr. Berlinski notes that “[a] Euclidean proof does not lend itself to light reading.”

He starts with Euclid’s first proposition, that on a given finite straight line, it is always possible to construct an equilateral triangle, again with commentary by later mathematicians and though a few mathematicians point out flaws, “when all is said and done, Euclid’s proof does what a proof must do: it compels belief.”

In Euclid’s 27th proposition, Euclid intentionally assumes a statement to be false then follows the chain of logic to end in a contradiction, thus proving the initial assumption to be false. Dr. Berlinski says this of Euclid’s method, “[t]he technique is known as reductio ad absurdum, or proof by contradiction. Euclid’s strategy is to prove that a proposition is true by assuming it is false, and then demonstrating what a mess it makes.”

Proposition 47 is the Pythagorean theorem, which is explained by way of modern algebra, something not available to Euclid. Dr. Berlinski notes, “Euclid’s proof of the Pythagorean theorem is therefore geometrical. There are no numbers, and no numbers are squared, but in his geometrical proof, Euclid found a way to convey arithmetical facts without mentioning them.”

Euclid limited his concrete constructions of abstract geometry to what could be done by straightedge and compass alone. This restriction prevents trisecting arbitrary angles, and the impossibility of proving the impossibility of squaring the circle. This had to wait for analytic geometry, and analytic geometry had to wait centuries after Euclid for the invention of the proper tools. Advances made by analytical geometry are touched on only lightly but is first posed as a chicken-and-egg question.

Which came first, (first as in the sense of more fundamental), geometry or arithmetic? Dr. Berlinski claims that Euclid is conflicted, evinced by the wording of his fifth axiom. Euclid is clumsy in expressing algebraic concepts in geometrical form. Though later mathematicians even with analytic geometry found some problems too hard to do. Proving that parallel lines never met posed such a difficulty that mathematicians in the nineteenth century gave up trying.

Analytic geometry came about through many mathematicians’ efforts over a long period of time. Rene Descartes provided the Euclidean coordinate system, Richard Dedekind added irrational numbers, and Ernst Steinitz provided the concept of the field.

In 1899 David Hilbert made an attempt to improve on Euclid by rewriting the Elements with a new axiom system to show geometry as a subset of real numbers. “Step by step Hilbert shows every one of the axioms in Euclidean geometry can be interpreted within a purely arithmetic model.” Dr. Berlinksi then hands the reader this wonderful simile. “He has gotten one theory to speak in another theory’s voice. It is like hearing a cat bark.”

Analytical geometry demonstrated its value to the modern world in 1916 through Einstein’s field equations for general relativity. Einstein’s great work represented a revolution in non-Euclidean thought, made possible by Gauss, Lobachevski, Bolyai, and Riemann.

Perhaps the publisher should produce a collector’s edition of The King of Infinite Space bound in leather with gold tooling for well-heeled readers to fill their mahogany paneled private libraries with tall bookshelves with wheeled ladders. Or for placement on Victorian tables, with nearby leather armchairs, antique brass telescopes and sepia-aged globes. And on that table, a book of poetry by Edna St. Vincent Millay, opened to Euclid Alone:

“Euclid alone has looked on Beauty bare. 

Let all who prate of Beauty hold their peace, 

And lay them prone upon the earth and cease 

To ponder on themselves, the while they stare 

At nothing, intricately drawn nowhere 

In shapes of shifting lineage; let geese 

Gabble and hiss, but heroes seek release 

From dusty bondage into luminous air. 

O blinding hour, O holy, terrible day, 

When first the shaft into his vision shone 

Of light anatomized! Euclid alone 

Has looked on Beauty bare. Fortunate they 

Who, though once only and then but far away, 

Have heard her massive sandal set on stone.”

The King of Infinite Space is for anyone who cares about Euclid, geometry, the philosophy of mathematics and, most especially, for those who appreciate fine writing.